- #1

Frillth

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## Homework Statement

Let S = {x∈R^n : x_i ≥ 0 for all i, x_1 + 2x_2 + 3x_3 + ... + nx_n ≤ n}. Find the n-dimensional volume of S.

## Homework Equations

I'm 95% sure that I'm supposed to use the change of variables theorem here.

## The Attempt at a Solution

So far, I have calculated the values for n from 1 to 6 using iterated integrals (calculated with mathematica). They are 1, 1, 3/4, 4/9, 125/576, and 9/100, respectively. I was hoping to get an easy pattern that I could use to lead me toward the general solution, but it looks like that won't be the case.

This is only the second COVT problem that we've been assigned, so I'm not quite comfortable with the theorem yet. I don't really have any idea what sort of change of variables would make this easy to deal with. Does anyone have any idea?

Edit: I just figured out the pattern based on some intuition and a lucky guess. The volume of S is (or seems to be, at least) n^n/(n!)^2. I have no idea how I could possibly get this by integration, however.

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